# Two weight inequalities for positive operators: doubling cubes

### Wei Chen

Yangzhou University, China and Georgia Institute of Technology, Atlanta, USA### Michael T. Lacey

Georgia Institute of Technology, Atlanta, USA

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## Abstract

For the maximal operator $M$ on $\mathbb{R}^{d}$, and $1 < p,\rho < \infty$, there is a finite constant $D=D _{p, \rho }$ so that this holds. For all weights $w,\sigma$ on $\mathbb{R}^{d}$, the operator $M(\sigma \cdot)$ is bounded from $L^{p}(\sigma )\to L^{p}(w)$ if and only if the pair of weights $(w,\sigma)$ satisfy the two weight $A_{p}$ condition, and this testing inequality holds:

for all cubes $Q$ for which there is a cube $P\supset Q$ satisfying $\sigma(P) \leq D\sigma(Q)$, and $\ell(P) \geq \rho \ell(Q)$. This was recently proved by Kangwei Li and Eric Sawyer. We give a short proof, which is easily seen to hold for several closely related operators.

## Cite this article

Wei Chen, Michael T. Lacey, Two weight inequalities for positive operators: doubling cubes. Rev. Mat. Iberoam. 36 (2020), no. 7, pp. 2209–2216

DOI 10.4171/RMI/1197